Related papers: On Those Analytic Functions of One Variable which …
This is a literal word-for-word translation from the French of Phragmen's proof (the first such published) of Weierstrass' famous theorem characterizing all analytic functions which possess an algebraic addition theorem.
We present a self-contained development of the Weierstrass theory of those analytic functions (single-valued or multiform) which admit an algebraic addition theorem. We review the history of the theory and present detailed proofs of the…
In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function $\wp(z)$ in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a…
The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in…
We prove that if an analytic map $f:=(f_1,\ldots ,f_n):U\subset \mathbb{C}^n\rightarrow \mathbb{C}^n$ admits an algebraic addition theorem then there exists a meromorphic map $g:=(g_1,\ldots ,g_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$…
Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.
The general analytic solution to the functional equation $$ \phi_1(x+y)= { { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over { \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is…
This is a systematic accounting of the classical theorems of Jordan and Tonelli, as well as an introduction to the theory of the Weierstrass integral which in its definitive form is due to Cesari. This is installment II of a four part…
We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…
We introduce the notion of the generalized-analytical function of the poly-number variable, which is a non-trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in…
Final representation of all those measures $\mu$ for which algebraic polynomials are dense in $L_p(R, d\mu)$ is found. The weighted analogue of the Weierstrass polynomial approximation theorem and a new version of the M. Krein's theorem…
Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq…
An technically interesting proof of a known theorem.
We present a new method to explicitly define Abelian functions associated with algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the procedure with the functions associated…
An algebraic technique adapted to the problems of the fundamental theoretical physics is presented. The exposition is an elaboration and an extension of the methods proposed in previous works by the aut
This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property,…
In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The…
A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity…
We construct the complex tangent as a meromorphic function in the plane, using an approach developed by Weierstrass in his characterization of analytic functions that satisfy algebraic addition theorems.
The validity of the Addition Theorem for algebraic entropies $\ent_L$ induced by non-discrete length functions $L$ on the category of locally $L$-finite modules over arbitrary rings is proved. Concrete examples of non-discrete length…